Semi‐Arithmetic Fuchsian Groups and Modular Embeddings
Semi‐Arithmetic Fuchsian Groups and Modular Embeddings
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半算术 Fuchsian 群和模块化嵌入
DOI：
10.1112/s0024610799008315
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
J. Wolfart
中科院分区：
文献类型：
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作者：
P. Schmutz Schaller;J. Wolfart
算术Fuchsian群是最有趣和最重要的Fuchsian群，因为它们在数论中的意义和它们的几何性质。然而，对于一个固定签名，只存在有限个非共轭算术Fuchsian群；因此，扩充这类富克斯族是可取的。这就是我们定义半算术Fuchsian群的动机。这样的群体可以定义如下（精确的表述见第2节）。设Γ是一个有限的Fuchsian群，设Γ2是由Γ的元素的平方生成的子群。如果Γ包含在作用于上半平面积Hr的算术群Δ中，则Γ是半算术群。同样，如果Γ2的所有元素的迹都是全实数域的代数整数，则Γ是半算术的。半算数的Fuchsian群的著名例子是三角群（及其有限指数的子群），除了Takeuchi[16]列出的85个三角群外，它们几乎都是非算数的。
Arithmetic Fuchsian groups are the most interesting and most important Fuchsian groups owing to their significance for number theory and owing to their geometric properties. However, for a fixed signature there exist only finitely many non‐conjugate arithmetic Fuchsian groups; it is therefore desirable to extend this class of Fuchsian groups. This is the motivation of our definition of semi‐arithmetic Fuchsian groups. Such a group may be defined as follows (for the precise formulation see Section 2). Let Γ be a cofinite Fuchsian group and let Γ2 be the subgroup generated by the squares of the elements of Γ. Then Γ is semi‐arithmetic if Γ is contained in an arithmetic group Δ acting on a product Hr of upper halfplanes. Equivalently, Γ is semi‐arithmetic if all traces of elements of Γ2 are algebraic integers of a totally real field. Well‐known examples of semi‐arithmetic Fuchsian groups are the triangle groups (and their subgroups of finite index) which are almost all non‐arithmetic with the exception of 85 triangle groups listed by Takeuchi [16].